Fermat's Last Theorem has been a huge debate until Wiles managed to convince the math community that he's made it... I know that a respected number of well achieved mathematicians, philosophers of mathematics and other intelligent people absolutely disagree with the "proof" by Andrew Wiles. Marilyn Vos Savant herself has written against that -so called- proof. The thing is that Wiles pulled the community into his "water" (which is made by him) and I absolutely know he's managed to trap everyone, leaving no chance for dispute. But that does not go for me!!It is all a matter of standing on the true view-point so that you see and manage to prove that Wiles is a math-trap...!I was nominated for the Abel Prize 2016, and had I received it, Wiles would be now known for having written a false proof. My published work is published and free-accessed by Saita Publications:http://www.saitabooks.eu/2015/01/ebook.134.htmlIn order to have a first impression on how I mean what I write, one has to consider that my work constitutes the proven Foundations of Mathematics. The wrong ones I have proven wrong and have replaced them, and the true ones I have actually proven!!! Even the ancient-long problem of the measurement of the circle has been acknowledged by Professor Doron Zeilberger (Euler Medal) as having been solved by me:http://www.math.rutgers.edu/~zeilberg/quotes.html(the 9th quote in his list)http://www.math.rutgers.edu/~zeilberg/khaver.html(nearly at the end of his list)I have posted a charge against the wrong FLT and Wiles' proof in the SORITES scholarly journal:http://sorites.org/room/index.htmDoes anyone think my work is worth the Abel Prize or other award, please give a hand! Thanks.

That's very interesting also for my studies as physicist.In your introduction of your book "The Groundlessness of Infinity" you wrote: "This leads to the abolition of the numeric set of any kind, and subsequently to the abolition of the numeric continuum and infinity".

So, if you believe that the notion of "continuum" is to be abolished, I understand that you coherently should oppose also "space-time continuum" as in Minkowski-Einstein relativity, and also as a keystone in GR (General Relativity), am I right?

As a Quantum mechanics physicist I'm totally opposing any possibility of "reconciliation" between relativism (especially that of GR) - which is based on the false concept of "space-time continuum" -and Quantum mechanics which is entirely revolving around the correct concept of DISCRETE in physical entities.

Your approach is very interesting. As you may read in unit 10 of my book, pure numbers not being connected in any way (i.e. absolutely discrete), leads to the logic that, e.g., 5 meters (as a non-pure numeric quantity) is a continuum (i.e. connection), but indeed a discrete continuum. This means that for any other quantity of length other that 5 meters, we refer to in the absolutely discrete way that is we cannot say "any quantity of meters" or "the length from 1 meter to 5 meters" as considering "any length" from 1 meter to 5 meters. So, pure numbers are not connected; and natural (cosmic) entities and magnitudes are connected as referred discretely (with unique reference as to their quantity). 7 electrons is a set (sum), but to refer to a subset, e.g. 2 electrons, you have to actually and solely refer to it and not leave it to vagueness; or else you haven't considered 2 electrons. You can also refer to an uncounted number (x) and state, e.g. x electrons<5 electrons. No matter in what way you consider that set, having referred solely to it you have actually considered it, that is "x electrons".

The same goes for macrocosmic entities, as I mentioned, and the essential thing to have in mind is that this logic springs from the notion and concept of the pure quantity that is number. It is not in my knowledge to guarantee that the macrocosm is constructed by the microcosm, but I've proven the uniqueness in quantitative (arithmetic) reference, and this is universal, apparently for both the microcosm and the macrocosm.

I suppose you'll find it very interesting to see how Zeno's Paradoxes are clearly solved by the logic of absolute discreteness in considering the quantity (unit 11 in my book).

Hi Emmanuel!And thanks for your answer.But I couldn’t find your book online and the sections you told me, I only could read your introduction.However, it seems to me that your solution – as far as you are explaining it to me - of discreteness is totally correct, no problem for that.My only doubt is about the “abolition of infinity”, because in my opinion the problem is more “formal” and “terminological” than else.But, as I’m a physicist, maybe I’m not able to catch some “subtle” concepts and “nuances” that you mathematicians are investigating more deeply.For instance, in my paper that I published last week on academia.edu https://www.academia.edu/27852699/Gravi ... stributionIf you read my section 13, where I’m proposing to see the Big Bang and the following expansion of the universe as a Dirac Delta function/distribution, I discussed the physical/mathematical problem of division of a physical entity/constant by 0.A colleague of mine (Stephen Crothers) complained that the famous astrophysicists Penrose and Hawking – in several of their papers - were dividing some physical magnitudes (such as mass) by 0, leading to infinity, because – Crothers said – it is impossible to divide a constant by 0, 1/0 does not yield infinity, but it is meaningless/indeterminate in mathematical terms.This of course is correct in the Real numbers domain, but it is allowed in the domain of HYPERREAL NUMBERS:That’s what I wrote:

“Furthermore, for a correct physical/mathematical approach, it is essential to point out that here we are using the concepts of “zero” and “infinity” in the sense of hyperreal numbers. [79]Infinity is not a number, in the real numbers field, but it is a number in the system of hyperreal numbers, or nonstandard real numbers. In this system, dividing by infinity gives infinitesimal (ε), which is not 0, but is closer to 0 than any real number.And therefore, dividing by 0 (which is not allowed in the field of real numbers, giving an undefined result) gives infinite (ω), which is not infinity, but is closer to infinity than any real number. Thus: 1/∞ = ε → 0 1/0 = ω → ∞ so that: 1/ ε = ω/1This extension of real numbers to hyperreal, as devised by Robinson and other mathematicians in the ’60, is fundamental in order to hold all the concepts of distributions and Delta δ also in physics, avoiding also the trouble of normalization of infinities, etc., as those so badly “hated” by Dirac.”

So, it seems to me that this is – more than anything else – a FORMAL, and not a physical problem, at least for us (physicists).In other words: you’re right when you say that we have to eliminate the concept of continuity, but I don’t believe we should eliminate infinity , PROVIDED we have clear in our minds what we mean with infinity and infinite.We cannot handle infinity as a number, that’s OK, but we can handle it as a DISCRETE approximation towards a limit, in the original meaning as expressed by Leibniz, and in the hyperreal numbers domain, i.e. a quantity that IS CLOSER TO INFINITY than anything else.This way we can “reconcile” the notion of DISCRETENESS – as you proposed in your book and papers – with that of infinity.In other words, hyperreal numbers propose a notion of infinity that is not a set number, it is a limit, and you can always imagine a DISCRETE gap between ω → ∞ and an actual infinity, or better, you can always (and more easily) imagine a DISCRETE GAP between ε → 0 and zero.I think this is also your approach with Zeno’s paradox, I’m I right?And, with respect to infinity it seems to me that this can solve also the famous “Hilbert hotel” paradox, don’t you believe ? (a next room that you can always add to infinite rooms, whenever a new guest is looking for accommodation)BestAlberto Miatello

Is there a link which does not require a person to accept a user agreement?

with respect to infinity it seems to me that this can solve also the famous “Hilbert hotel” paradox, don’t you believe ? (a next room that you can always add to infinite rooms, whenever a new guest is looking for accommodation)

Assume Hilbert's hotel is linear. The message is sent to guests to move into the next higher numbered room. However, this message can travel no faster than the speed of light. Thus at a time T some people have moved, others are in the process of moving, others have not yet received the message. If the hotel is assumed full then at no time have all guests moved to the next room because this must take infinite time. If the hotel is populated by this method then at no time is the hotel full because it takes infinite time to accomplish this- it is always being filled, but is never full. Thus we see that the Hilbert Hotel problem vanishes because it is a non-real-world fantasy.

I should firstly explain that I do not abolish the concept of finite continuity of non-numeric entities (that is anything other than pure numbers) in any way; from 1 meter to 5 meters there is the distance of 4 meters. But "4 meters" has to be discretely mentioned (referred or counted) for in my work the concept of abstract (vague) reference (counting) is the wrong one, and not the concept of any specific continuum.

As for hyperreal numbers and zero, I assure you'll get the answer once you see that the only true reference in numbers is the unique one (one number at a reference/measurement). Therefrore in terms of "The Groundlessnes of Infinity", a number and a function of measurement exists only if we use the absolutely discrete (unique, i.e. one-at-a-time) counting. So approximation does not belong in arithmetic, but rather in equations using notions other than numbers, and that is any descriptive/constructive way which does not belong in arithmetic. But when numbers are involved in a construction (equation or function) then one has to respect the discreteness of the numeric agents. More simply, when combining a number with another (physical) notion, the number requires that it is discretely treated.

As fot the need to use the notion of infinity, please see in the book how I mean what I write in the description: "...it is the same logic that abolishes infinity, which abolishes the notion of a supposed "ultimate end", by introducing a new way of regarding the "limit" in counting."

The issue of infinity genuinely belongs to the concept of the numeric continuum. When working with concepts that don't involve numbers, as in the concept of speed (u=s/t) or any other that is irrelevant of counting, then the notion of infinity is also irrelevant. But when we want to see if speed can be infinite, then we do not deal with the function of the concept "speed", but with the number of it, i.e. if in n(km/h) n can be infinite.

So, infinity is abolished by the proof that there is no numeric continuum, as much as the finite continuum "n meters" is abolished when n is not a specific (uniquely referred) number (it doesn't matter if it is of known or unknown magnitude).

In conclusion, the concept of approximation is meaningfull only in the cases where the approximation is not arithmetic approximation, but a function that does not rely on numbers as to its approximation. And it is mostly interesting to see the logic by which irrational numbers and infinite decimals cannot be, in unit 5. In unit 6 you can read about the approximation in numbers.

[*]Is there a link which does not require a person to accept a user agreement? "

But you don't need to accept a "user agreement". You can open a personal page of a user of academia.edu - while staying anonymous - and then you can click the paper you want to read, and you find the label "Read" in the page, and then you can scroll all the pages.You need either to register, or to disclose your indentity ONLY when you want to DOWNLOAD a paper. But if you want to read only, you may stay Anonymous.

[*] However, this message can travel no faster than the speed of light. Thus at a time T some people have moved, others are in the process of moving, others have not yet received the message... "

No, this does not seem the correct answer. Grand Hotel Hilbert is a "thought" MATHEMATICAL paradox. As such it is forbidden to introduce PHYSICAL explanations as yours. Otherwise one could raise the - even stronger - objection that a new guest -traveling at "just" the speed of light - should need INFINITE time just to know whether the last room is empty or not, when he/she arrives to the reception.So, the Grand Hotel Hilbert is just a mere Mathematical model to explain why transfinite numbers are a sub-set possessing cardinality, i.e. you can associate in a bijective function all infinite rooms with its subset of odd-numbered rooms, and that does not happen with sets possessing just finite elements.

Albert 2016 wrote:..you find the label READ and you can read all the 76 pages, one by one.

I missed that. Prefer to read offline, but this will suffice.

Robert 46 wrote: However, this message can travel no faster than the speed of light. Thus at a time T some people have moved, others are in the process of moving, others have not yet received the message... "

No, this does not seem the correct answer. Grand Hotel Hilbert is a "thought" MATHEMATICAL paradox. As such it is forbidden to introduce PHYSICAL explanations as yours.

What this means is that there is a disconnect between the fantasy realm of mathematics and the real world. I doubt that mathematics can tell us anything exact about the universe. Yet, dimensional analysis is a powerful tool to tell us what the form of a physical equation must be. However, any factor where the units of mass/length/time cancel out could be valid, and so too any term having the same dimensions as another term. Mathematics does not point us toward the discovery of such other factors and terms.

So, the Grand Hotel Hilbert is just a mere Mathematical model to explain why transfinite numbers are a sub-set possessing cardinality, i.e. you can associate in a bijective function all infinite rooms with its subset of odd-numbered rooms, and that does not happen with sets possessing just finite elements.

Yet the entire realm of mathematical infinity introduces the non-real-world concept that a part can be as large as the whole- which is contradicted by everything natural which we would consider in comparison.

I absolutely agree on your last comment; infinity does not allow reality to actually exist; infinity is an abstract so-called "concept", but how is the concept which is absolutely selfish and ignoring all real concepts allowed to be in our minds?? It is a matter of intellectual dignity, and we have to be able to get rid of "infinity". I claim to have abolished it, and anyone is free to realise that. I think it's a good thing that my book is free on the internet.

You also commented on the difference of Physics with Mathematics. Yet, after my book was published by Saita Publications, I added the (purely mathematical) proof on Einstein's gravitational field. This edition is available by Amazon, and one can find it by clicking on the words "...Amazon version..." of the last paragraph of the short description:

Emmanuel Xagorarakis wrote:But you're not afraid of me, let alone my work... So, do you think you could first study it and then give RESPONSIBLE evaluation? I think you won't do that, but anyway, thanks for your kindness.

As I promised, I’m sending you my comment of your book.Your book is very interesting, and there are several ideas that – in my opinion – are correct.For instance, I’d like to quote what you write at p. 60“About the paradox of the Arrow [23], the abolition of the abstract continuum iswhat saves the concept of the continuum itself. The arrow being still at any momentof its course (because the moment has no magnitude) is a wrong statement:referring to any moment means the abstract reference to the time (the moments) ofthe movement of the arrow. The abolition of the abstract continuum directs tospecific and discrete time references. So, in the moment A of the arrow’s course thedistance covered will be a specific distance Ad. For a different moment B, thedistance is different; Bd. This secures that the arrow actually covers distance andthus it is moving.”

Of course I fully 100% agree with you, also because I’m a quantum mechanics physicist, and DISCRETENESS in energy transmission is one of the main cornerstones of quantum mechanics.My only doubt is not regarding what you write (I could not read everything of your book, in 2 days I just selected some sections, and pages, but again, your ideas seem correct to me).

My only doubt is about THE WAY you present your ideas.I noted that in your work there is no reference, and no bibliography.Why?Unless someone is presenting a totally new research – for instance a study on a new source of energy, a new mechanical, electronic, device, a new invention, prototype, etc. – but the field of your investigation (infinity in mathematics) is one of the MOST investigated fields of mathematics, for almost 25 centuries.

If we look for “infinity” in Mathematics, in a bibliotheca whatever, we can find books from Cantor, Zermelo, Weirstrass, Leibniz, Gödel, Hilbert, Hamilton, etc. etc.

So, if you write something about infinity in mathematics, and you don’t compare your ideas with those of major authors above, that is immediately making “suspicious” your research.Because what you write is not at all new, so you have to tell us WHY and WHERE what you write is different from that those authors have already written .

And if you don’t mention those authors, it is like admitting that you did not read them, and that is really a very BAD idea.

No university would accept a study and scientific research having no bibliography.